<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<title></title>
<!--Generated on Tue Jun  3 20:03:23 2014 by LaTeXML (version 0.7.999_04) http://dlmf.nist.gov/LaTeXML/.-->

<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link rel="stylesheet" href="../../../../../../../LaTeXML.css" type="text/css">
<link rel="stylesheet" href="../../../../../../../ltx-article.css" type="text/css">
<link rel="stylesheet" href="../../../../../../../customRules.css" type="text/css">
</head>
<body>
<div class="ltx_page_main">
<div class="ltx_page_content">
<div class="ltx_document">
<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Radial symmetry</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">This algorithm finds the sub-pixel position of a molecule by determining
the point with maximal radial symmetry in the data as described in
<cite class="ltx_cite">[<a href="#bib.bib24" title="Rapid, accurate particle tracking by calculation of radial symmetry centers" class="ltx_ref">1</a>]</cite>. The general idea is to find the origin
of radial symmetry (i.e., the center of a molecule) as the point with
the minimum distance to gradient-oriented lines passing through all
data points. The calculation of each molecular position is very fast
due to an analytical solution, but the algorithm does not estimate
the intensity or imaged size of a molecule. Radial symmetry is a robust
feature in SMLM data, making the algorithm resistant to noise.</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">The calculation starts by determining the intensity co-gradient</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-24px" src="mi/mi129.png" width="245" height="61" alt="\nabla\tilde{I}_{x,y}=\left[\frac{\partial\tilde{I}\left(x,y\right)}{\partial u%
},\frac{\partial\tilde{I}\left(x,y\right)}{\partial v}\right]^{\top}"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">for every integer point <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi60.png" width="45" height="21" alt="\left(x,y\right)"> from the set <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi37.png" width="18" height="16" alt="\mathcal{D}">.
Here <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi149.png" width="23" height="12" alt="uv"> coordinates are rotated by <img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi135.png" width="30" height="16" alt="45^{\circ}"> from the <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi69.png" width="23" height="15" alt="xy">
coordinate system of the image, because partial derivatives are determined
using the Roberts cross operator <cite class="ltx_cite">[<a href="#bib.bib1" title="Image Processing, Analysis, and Machine Vision" class="ltx_ref">2</a>]</cite> as</p>
<table id="Sx1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.Ex1" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-14px" src="mi/mi121.png" width="72" height="44" alt="\displaystyle\frac{\partial\tilde{I}\left(x,y\right)}{\partial u}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi130.png" width="203" height="24" alt="\displaystyle\tilde{I}\left(x+1,y+1\right)-\tilde{I}\left(x,y\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
<tr id="Sx1.Ex2" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex2.m1" class="ltx_Math" style="vertical-align:-14px" src="mi/mi123.png" width="72" height="44" alt="\displaystyle\frac{\partial\tilde{I}\left(x,y\right)}{\partial v}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex2.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi131.png" width="203" height="24" alt="\displaystyle\tilde{I}\left(x,y+1\right)-\tilde{I}\left(x+1,y\right)\,."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
</div>
<div id="Sx1.p3" class="ltx_para">
<p class="ltx_p">The computed co-gradient <img id="Sx1.p3.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi141.png" width="47" height="25" alt="\nabla\tilde{I}_{x,y}"> corresponds to the
point <img id="Sx1.p3.m2" class="ltx_Math" style="vertical-align:-7px" src="mi/mi136.png" width="190" height="22" alt="A_{x,y}=\left(x+0.5,y+0.5\right)"> and the slope of a gradient-oriented
line passing through the point <img id="Sx1.p3.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi137.png" width="38" height="21" alt="A_{x,y}"> is, in a <img id="Sx1.p3.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi69.png" width="23" height="15" alt="xy"> coordinate
system, given by</p>
<table id="Sx1.E2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi128.png" width="453" height="50" alt="\mathit{s_{x,y}}=\left(\frac{\partial I\left(x,y\right)}{\partial u}+\frac{%
\partial I\left(x,y\right)}{\partial v}\right)\left(\frac{\partial I\left(x,y%
\right)}{\partial u}-\frac{\partial I\left(x,y\right)}{\partial v}\right)^{-1}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
</div>
<div id="Sx1.p4" class="ltx_para">
<p class="ltx_p">The origin of the radial symmetry <img id="Sx1.p4.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi140.png" width="60" height="21" alt="\left(\hat{x}_{0},\hat{y}_{0}\right)">
can be determined as the point that minimizes the sum of weighed distances
of all considered lines to that point. It can be shown <cite class="ltx_cite">[<a href="#bib.bib24" title="Rapid, accurate particle tracking by calculation of radial symmetry centers" class="ltx_ref">1</a>]</cite>
that the problem has an analytical solution. For simplicity, let index
<img id="Sx1.p4.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi147.png" width="141" height="21" alt="j=x\left(2r+1\right)+y">, where <img id="Sx1.p4.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi152.png" width="67" height="19" alt="x,y\in\mathcal{D}">, thus <img id="Sx1.p4.m4" class="ltx_Math" style="vertical-align:-7px" src="mi/mi148.png" width="72" height="17" alt="s_{j}=s_{x,y}\,">,
<img id="Sx1.p4.m5" class="ltx_Math" style="vertical-align:-7px" src="mi/mi151.png" width="81" height="17" alt="w_{j}=w_{x,y}\,">, and <img id="Sx1.p4.m6" class="ltx_Math" style="vertical-align:-7px" src="mi/mi144.png" width="71" height="21" alt="b_{j}=b_{x,y}\,">. The analytical solution
is given by the following equations</p>
<table id="Sx1.EGx2" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.E3" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E3.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi125.png" width="22" height="19" alt="\displaystyle\hat{x}_{0}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E3.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E3.m3" class="ltx_Math" style="vertical-align:-27px" src="mi/mi133.png" width="426" height="54" alt="\displaystyle a^{-1}\biggl[\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}b_{j}}\Bigr)%
\Bigl(\sum_{j\in\mathcal{D}}{w_{j}}\Bigr)-\Bigl(\sum_{j\in\mathcal{D}}{w_{j}b_%
{j}}\Bigr)\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}}\Bigr)\biggr]\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(3)</span></td>
</tr>
<tr id="Sx1.E4" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.E4.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi127.png" width="21" height="19" alt="\displaystyle\hat{y}_{0}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.E4.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.E4.m3" class="ltx_Math" style="vertical-align:-27px" src="mi/mi132.png" width="442" height="54" alt="\displaystyle a^{-1}\biggl[\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}b_{j}}\Bigr)%
\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}}\Bigr)-\Bigl(\sum_{j\in\mathcal{D}}{w_%
{j}b_{j}}\Bigr)\Bigl(\sum_{j\in\mathcal{D}}{s_{j}^{2}w_{j}}\Bigr)\biggr]\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(4)</span></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.p4.m7" class="ltx_Math" style="vertical-align:-10px" src="mi/mi143.png" width="355" height="30" alt="a=\bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}}\bigr)^{2}-\bigl(\sum_{j\in\mathcal{%
D}}{s_{j}^{2}w_{j}}\bigr)\bigl(\sum_{j\in\mathcal{D}}{w_{j}}\bigr)\,">,
<img id="Sx1.p4.m8" class="ltx_Math" style="vertical-align:-7px" src="mi/mi145.png" width="99" height="21" alt="b_{j}=y-s_{j}x\,">, and <img id="Sx1.p4.m9" class="ltx_Math" style="vertical-align:-15px" src="mi/mi150.png" width="83" height="34" alt="w_{j}=\frac{\tilde{w}_{j}}{s_{j}^{2}+1}\,">.
The point-to-line distances are weighted by the factor <img id="Sx1.p4.m10" class="ltx_Math" style="vertical-align:-20px" src="mi/mi142.png" width="176" height="49" alt="\tilde{w}_{j}=\tilde{w}_{x,y}=\frac{\bigl|\nabla\tilde{I}_{x,y}\bigr|^{2}}{d%
\bigl(A_{x,y},C\bigr)}\,">,
where <img id="Sx1.p4.m11" class="ltx_Math" style="vertical-align:-8px" src="mi/mi146.png" width="85" height="25" alt="d\bigl(A_{x,y},C\bigr)"> is the Euclidean distance of point
<img id="Sx1.p4.m12" class="ltx_Math" style="vertical-align:-7px" src="mi/mi137.png" width="38" height="21" alt="A_{x,y}"> to the centroid <img id="Sx1.p4.m13" class="ltx_Math" style="vertical-align:-2px" src="mi/mi138.png" width="18" height="16" alt="C">. The centroid is computed analogously
to <a href="CenterOfMassEstimatorUI.html" title="" class="ltx_ref">centroid estimator</a> but from
the gradient magnitudes <img id="Sx1.p4.m14" class="ltx_Math" style="vertical-align:-8px" src="mi/mi139.png" width="59" height="26" alt="\bigl|\nabla\tilde{I}_{x,y}\bigr|">.</p>
</div>
<div id="Sx1.p5" class="ltx_para">
<p class="ltx_p">The implementation of the radial symmetry method is taken from <cite class="ltx_cite">[<a href="#bib.bib24" title="Rapid, accurate particle tracking by calculation of radial symmetry centers" class="ltx_ref">1</a>]</cite>,
where the partial derivatives are additionally smoothed by a <img id="Sx1.p5.m1" class="ltx_Math" style="vertical-align:-3px" src="mi/mi134.png" width="43" height="17" alt="3\times 3">
averaging filter, see Section <span class="ltx_ref ltx_ref_self">LABEL:sub:Averaging-filter</span>, in order
to reduce noise and improve the accuracy of the results.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Estimators.html" title="" class="ltx_ref">Sub-pixel 2D localization of molecules</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib24" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">R. Parthasarathy</span><span class="ltx_text ltx_bib_year">(2012)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Rapid, accurate particle tracking by calculation of radial symmetry centers</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Nature Methods</span> <span class="ltx_text ltx_bib_volume">9</span> (<span class="ltx_text ltx_bib_number">7</span>), <span class="ltx_text ltx_bib_pages"> pp. 724–6</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1038/nmeth.2071" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Radial symmetry" class="ltx_ref"><span class="ltx_text ltx_ref_title">Radial symmetry</span></a>,
<a href="#Sx1.p4" title="Radial symmetry" class="ltx_ref"><span class="ltx_text ltx_ref_title">Radial symmetry</span></a>,
<a href="#Sx1.p5" title="Radial symmetry" class="ltx_ref"><span class="ltx_text ltx_ref_title">Radial symmetry</span></a>.
</span>
</li>
<li id="bib.bib1" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">M. Šonka, V. Hlaváč and R. Boyle</span><span class="ltx_text ltx_bib_year">(2007)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Image Processing, Analysis, and Machine Vision</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_edition">3rd edition edition</span>,  <span class="ltx_text ltx_bib_publisher">Cengage Learning</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p2" title="Radial symmetry" class="ltx_ref"><span class="ltx_text ltx_ref_title">Radial symmetry</span></a>.
</span>
</li>
</ul>
</div>
</div>
</div>
<div class="ltx_page_footer">
<div class="ltx_page_logo">Generated  on Tue Jun  3 20:03:23 2014 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="" alt="[LOGO]"></a>
</div>
</div>
</div>
</body>
</html>
